Let the variables Xi(i = 1,...,s) be independently distributed with Poisson distribution P(i). For testing the hypothesis

Let the variables Xi(i = 1,...,s) be independently distributed with Poisson distribution P(λi). For testing the hypothesis H :

λj ≤ a (for example, that the combined radioactivity of a number of pieces of radioactive material does not exceed a), there exists a UMP test, which rejects when  X j > C.



[If the joint distribution of the X’s is factored into the marginal distribution of X j (Poisson with mean λj) times the conditional distribution of the variables Yi = X j /

 X j given  X j (multinomial with probabilities pi = λi/

λj), the argument is analogous to that given in Example 3.8.1.]